Stretching, Compressing, and Reflecting Graphs of Functions
Big Idea: We can transform any function $y = f(x)$ by replacing $x$ with expressions like $-x$ or $kx$, or by negating the entire function. Each algebraic change corresponds to a geometric transformation of the graph.
1. Reflection About the Y-Axis: $y = f(-x)$
The graph of $y = f(-x)$ is obtained from the graph of $y = f(x)$ by reflecting every point across the y-axis.
Rule: A point $(x_1, y_1)$ on $y = f(x)$ corresponds to the point $(-x_1, y_1)$ on $y = f(-x)$.
Think of it this way: the y-coordinate stays the same, but the x-coordinate flips sign. The graph becomes a mirror image in the y-axis.
Example: To graph $y = \sqrt{-x - 1}$, we recognize this as $y = \sqrt{-(x+1)}$, which is $y = \sqrt{x-1}$ reflected about the y-axis and then shifted. We'll work through this in detail below.
Interactive visualization — available in the full lesson